Lecture 1. Introductio n
1.1 Th e grou p theoreti c settin g fro m whic h ou r subjec t aros e i s th e following . W e
are give n a grou p G an d conside r a shor t exac t sequenc e
(1) I-+R
F
^ G - + 1,
where F i s a fre e group . Thi s i s calle d a (free) presentation o f G. Le t * j , x2, . . .
freely generat e F an d w^, w^ . . . b e generator s o f R a s a norma l subgrou p o f F . W e
shall writ e
F = (x
v
%
2
, . . . ) ,
and
R = ( F
( W J , W
V
. . . ) .
Then7 7 yields a definitio n o f G i n term s o f generators and relations:
G = (gl g
2
' ' * " ; w lW = ^ 2^ = = 1),
whereg^ . = xx an d ^ (g ) = w.n. Conversely , suc h a definitio n o f G yield s a fre e pres -
entation: w e simpl y tak e F t o b e th e fre e grou p o n element s %. , *
2
, . . . an d 7 7 th e
group homomorphis m extendin g x. \~+ g., i = 1, 2 , . . . . The n w.(x) e R fo r al l ; an d
R mus t b e th e norma l closur e o f al l th e w.(x).
R i s ofte n calle d th e relation group determine d b y 77 , or b y (1).
1 .2 I f th e numbe r o f th e x. an d th e w. i s finite , the n (1) i s calle d a finite pres-
entation an d G i s called finitely presentable. I n particular, ever y finit e grou p i s fi -
nitely presentable .
In thes e lecture s w e shal l b e considerin g onl y finit e group s G . Withou t an y essen -
tial los s o f generality , w e ca n therefor e restric t ou r fre e group s F t o b e finitel y gen -
erated. Th e relatio n group s R wil l the n als o b e finitel y generated . Indeed , b y Schreier' s
theorem, it d(F) i s th e minimu m numbe r o f generator s o f F , the n th e correspondin g
number fo r R is give n b y th e formul a d(R) = \G\(d(F) - l ) + 1.
1 . 3 Th e presentatio n theor y o f a fixed grou p G is concerne d wit h comparin g it s
possible presentations .
For example , suppos e 77. : F - » G, 2 = 1 , 2 , ar e tw o presentations , involvin g th e
same fre e grou p F . Ho w ar e the y related ? A ver y stron g connexio n woul d b e th e ex -
istence o f automorphism s 6 o f F an d f o f G s o tha t n^ = On-. Th e stud y o f thi s
1
http://dx.doi.org/10.1090/cbms/025/01
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